One Dimensional Oscillator (small oscillations)

• Consider a system with one degree of freedom and one generalized co-ordinate q. For small displacement from the equilibrium we can expand potential energy function using taylor series expansion about the equilibrium and we will only consider the lowest order terms. So expanding PE function V(q) we have
  
  
  
  
  
  
  where derivativesare evaluated at the equilibrium position q=q₀ and at equlilbrium (∂V/∂q)₀ = 0
• V(q₀) is potential energy at equilibrium and can be taken as zero, if the origin of potential energy is shifted to be at minimum equilibrium value.
  This implies that
  
  
  
  
  
  
  
  
  putting second derivative term in bracket equal to k and shifting origin to q₀=0 we have
  V(q)=kq²/2
  and k is the positive parameter at the position of stable equilibrium.
• If generalized co-ordinates does not involve time explicitely , the K.E. is then homogeneous quadratic function of generalized velocities, or,.
  
  

where coefficent m(q) , is in general function of q co-ordinate and may also be expanded in taylor series about the equilibrium position as we have done for potential energy function and first derivative of q is quadratic in this equation and the lowest lowest nonvanishing approximation to T is obtained by retaining only the firt term in the Taylor series expansion of m(q) which is m(0). Thus Lagrangian for small oscillations of one dimensional oscillator is

and equation of motion is